
In the most general case, \(f\)  is succinctly described by an arithmetic circuitFormally we need to require some restrictions on this arithmetic circuit to avoid doubly-exponentially large numbers by repeated squaring, see [1]}, [2]} for details.

When defining the class , [3]} consider the case where \(f\)  is guaranteedFormally to avoid promise problems the algorithm may return a certificate that the component-wise convexity is violated. In contrast, for the fourth variant it is easy to tell if a monomial is component-wise convex. to be component-wise convex.

[3]} consider the restriction that \(f\)  is explicitly given as a sum of monomials (note that this is orthogonal to the component-wise convexity condition). This problem is called KKT in [3]}.

In the most restrictive case, \(f\)  is a sum of component-wise convex monomials of constant degree.

